How do I prune this plant?
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Your picture at first glance appeared to have two plants. It would make sense since it appears from the leaf structure that you have a rose plant that has not received care for some time. The flowers also appear to be producing rose hips. (seeds)
Modern roses are often one type of rose grafted onto a hearty root stock. If you look just above ground-level you can often see where there is a scar from where the original root stock was cut off and new canes grafted on.
Over time the root stock will send up shoots from below ground and the plant will start reverting back to the original root stock. One pruning task is to remove these shoots, called suckers, as they come up.
You can find an article on pruning roses here:
While the plant has attractive flowers, it is very overgrown and the new shoots contain some very aggressive thorns. The first question is do you want to try to rescue this plant or would another plant be better?
If all the canes appear to be coming up from the ground rather than off a central stem, the plant has gone back to its root stock and many gardeners would replace the plant at that point.
If you decide to keep the plants, a good place to start would be to remove the canes coming out of the ground away from the central plant. That can be done now. Major pruning should be done in February through March when the plant is dormant. For the remainder of the year, follow the pruning instructions in the article and you may be rewarded with more blooms.